Two dimensional elliptic equation with critical nonlinear growth

Abstract:

We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions. \begin{equation*}\tag {1.1} \begin {cases} -\Delta u= \lambda ue^{u^2}, u>0 & \text {in $\Omega $}, \\ u=0 & \text {on $\partial \Omega $}, \end{cases} \end{equation*} where $\Omega$ is a unit disk in $\mathbb {R}^2$ and $\lambda$ denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies \begin{equation*} \int _{D}\left \vert \nabla u\right \vert ^{2}dx\rightarrow 4\pi ,\quad \lambda \searrow 0. \end{equation*} Moreover, by using the Pohozaev identity to the rescaled equation, we show that for any finite energy radially symmetric solutions to (1.1), there is a rescaled asymptotics such as \begin{equation*} u_m^2(\gamma _m x)-u_m^2 (\gamma _m)\to 2\log \frac {2}{1+|x|^2} \quad \text {as }\lambda _m\searrow 0 \end{equation*} locally uniformly in $x\in \mathbb R^2$. We also show some extensions of the above results for general two dimensional domains.